Efectuar o deste Artigo (vers o PDF) MotaSoares-et-al-CMAMEng-1997

Efectuar o deste Artigo (vers o PDF) MotaSoares-et-al-CMAMEng-1997
Computer methods
in applied
mechanics and
engineering
Comput. Methods Appl. Mech.
ELSEYIER
Optimization
Engrg. 149 (1997)
133-152
of multilaminated structures using higher-order
deformation models
Crist&%o M. Mota Soares”‘“, Carlos A. Mota Soares”, Victor M. Franc0 Correiab
“IDMEC-Instiruto
de Engenharia
hENIDH-Escola
N&rim
Mecrinica-lnsriruto
lnfunir
Superior
D. Henriquu,
Tkxico,
Av. Rovisco Pais,
Av. Eng. Bormrville
France,
1096 Lishm
2780 Oeiras,
Codex, Portu,qcd
Portugcd
Abstract
A refined shear deformation theory assuming a non-linear variation for the displacement field is used to develop discrete models for the
sensitivity analysis and optimization
of thick and thin multilayered
angle ply composite plate structures. The structural and sensitivity
analysis formulation is developed for a family of C” Lagrangian elements, with eleven, nine and seven degrees of freedom per node using a
single layer formulation. The design sensitivities of structural response for static, free vibrations and buckling situations for objective and/or
constraint functions with respect to ply angles and ply thicknesses are developed. These different objectives and/or
generalized displacements at specified nodes, Hoffman’s
or the volume
constraints can be
stress failure criterion, elastic strain energy, natural frequencies of chosen vibration
modes, buckling
load parameter
semi-analytically.
The accuracy and relative performance of the proposed discrete models are compared and discussed among the developed
of structural material.
The design sensitivities are evaluated
elements and with alternative models. A few illustrative test designs are discussed to show the applicability
either analytically
or
of the proposed models.
1. Introduction
Laminated composite materials are being widely used in many industries mainly because they allow design
engineers to achieve very important weight reductions when compared to traditional materials and also because
more complex shapes can be easily obtained. The mechanical behavior of a laminate is strongly dependent of
the fiber directions and because of that the laminate should be designed to meet the specific requirements of
each particular application in order to obtain the maximum advantages of such materials. Accurate and efficient
structural analysis, design sensitivity analysis and optimization procedures are very important to accomplish this
task. Structural optimization with behavioral constraints, such as stress failure criterion, maximum deflection,
natural frequencies and buckling load can be very useful in significantly improving the performance of the
structures by manipulating certain design variables. Design sensitivity analysis is important to accurately know
the effects of design variables changes on the performance of structures by calculating the search directions to
find an optimum design. To evaluate these sensitivities
efficiently and accurately it is important to have
appropriate techniques associated to good structural models.
It is well known that the analysis of laminated composite structures by using the classical Kirchhoff
assumptions can lead to substantial errors for moderately thick plates or shells. This is mainly due to neglecting
the transverse shear deformation effects which become very important in composite materials with low ratios of
transverse shear modulus to in-plane modulus. This can be attenuated by Mindlin’s first-order shear deformation
theory [ 1,2], but this theory yields a constant shear strain variation through the thickness and therefore requires
the use of shear correction factors [3] in order to approximate the quadratic distribution in the elasticity theory.
* Corresponding author
0045.7825/97/$17.00
P/I
0
S0045-7825(97)00066-2
1997 Elsevier Science S.A. All rights reserved
134
C.M.M. Soar-es et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152
More accurate numerical models such as three-dimensional
finite elements models can be used with adequate
refined meshes in order to contemplate acceptable aspect ratios, but these models are computationally
expensive.
A compromising less expensive situation can be achieved by using single layer models, based on higher-order
displacement
fields involving higher-order expansions of the displacement
field in powers of the thickness
coordinate. These models can accurately account for the effects of transverse shear deformation yielding
quadratic variation of out-of-plane
strains and therefore do not require the use of artificial shear correction
factors and are suitable for the analysis of highly anisotropic plates ranging from high to low length-to-thickness
ratios.
Pioneering work on the structural analysis formulation based in higher-order displacement
fields can be
reviewed in [4,5] where a theory which accounts for the effects of transverse shear deformation, transverse
strain and nonlinear distribution of the in-plane displacements with respect to thickness coordinate is developed.
Third-order theories have been proposed by Reddy [6-81, Phan and Reddy [9], Librescu [lo], Schmidt [ 111,
Murty [ 121, Lenvinson [ 131, Seide [ 141, Murthy [ 151, Bhimaraddi and Stevens [ 161, Mallikarjuna and Kant [ 171
and Kant and Pandia [ 181 among others. Related overviews, closed form solutions and discrete models based on
higher-order displacements fields can be found in Reddy [6,8,19], Mallikarjuna and Kant [ 171, Noor and Burton
[20], Bert [21], Kant and Kommineni [22], Reddy and Robbins Jr. [23] and Robbins Jr. and Reddy [24] among
others. Phan and Reddy [9] and Reddy and Phan [25] used a higher-order shear deformation theory based in a
displacement field which accounts for layerwise parabolic distribution of transverse shear stress that satisfies the
stress-free boundary conditions at the top and bottom surfaces of laminated plates, applied to the calculation of
deflections, stresses, buckling loads and natural frequencies.
Closed form solutions for simply supported
laminated plates and a finite element model where the transverse displacement uses C’ continuity Hermite cubic
shape functions to assure the continuity of the deflection and its derivatives across interelement boundaries were
also analyzed. Other studies using the same displacement field are due to Reddy and Khdeir [26] and Khdeir
[27], where closed form solutions are obtained and compared with classical plate theory (CPT) and higher-order
shear deformation theories (HSDT) for unsymmetric cross ply rectangular composite laminates for buckling and
free vibration under several boundary conditions.
Finite element higher-order discrete models have been
developed and discussed for vibration and/or buckling by Senthilnathan et al. [28], Kant et al. 1291, Putcha and
Reddy [30], Kozma and Ochoa [31], Mallikarjuna and Kant [17], Liu [32], and Mallikarjuna and Kant [33].
Recently a four node rectangular element for the buckling analysis of multilaminated
plates has been
developed by Ghosh and Dey [34,35]. This model assumes a parabolic distribution of the transverse shear
stresses and the non-linearity of the in-plane displacements across the thickness. The geometric stiffness matrix
is developed using the in-plane stresses.
An analytical solution based on a local high order deformation theory used for the determination
of the
natural frequencies and buckling analysis of laminated plates has been proposed by Wu and Chen [36]. The
displacement fields in this theory are assumed to be piece-wise continuous high order polinomial series, layer by
layer or sublaminate by sublaminate in thickness direction, accounting for the effects of transverse shear and
normal deformation.
Moita et al. [37] presented an eight node isoparametric discrete model based on an
third-order expansion in the coordinate for the in-plane displacements and a constant transverse displacement.
The model is applied to study several cases of composite plate and shell structures taking into consideration
different number of layers, lamination
angles, length-to-thickness
ratios as well as symmetric and nonsymmetric laminates. The influence that the higher-order terms incorporated in the geometric stiffness matrix
have on the prediction of the buckling load is also discussed.
Related buckling studies are due to Kam and Chang [38] who developed a finite element plate model, based
on the first-order shear deformation theory (FSDT) in which shear correction factors are derived from the exact
expressions
for orthotropic materials. Comprehensive
overviews of buckling and vibration of composite
plate-shell structures are given by Noor and Peters [39], Leissa [40,41], Kapania and Raciti [42,43] and
Palazotto and Dennis [44]. Buckling analysis of moderately thick laminated plate and shell structures has been
recently reviewed by Simitses [45] reporting works based on higher-order shear deformation theory and/or
first-order shear deformation theory with or without a shear correction factor.
Literature surveys in the field of sensitivity analysis and structural optimization such as those of Adelman and
Haftka [46], Haftka and Adelman [47], Zyczowski [48] and Grandhi [49] reveal some lack of studies carried out
on composite structures. Recently, Abrate [50] gave a wide perspective of work carried out by different
researchers in the field of the optimum design of composite laminated plates and shells subjected to constraints
on strength, stiffness, buckling loads and fundamental natural frequencies. Of the 84 papers reviewed, most of
C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg.
149 (1997) 1.3.3-152
135
them are based on variational approximation methods and the use of higher-order models is not mentioned. The
optimal design of laminated plates for maximum buckling using discrete ply angles and a multiobjective
approach to determine the optimal stacking sequence has been recently presented by Adali et al. [51,52].
Higher-order models have been used on the identification of material properties of multilaminated
specimens
using sensitivity analysis, optimization techniques and experimental vibration data enabling the identification of
six mechanical properties [53-561. Csonka et al. [57] developed a higher-order axisymmetric discrete model
where the nonlinear order theory is developed for the meridional displacement component through the thickness
of the axisymmetric shell. The radial coordinate of a nodal point, ply thickness and the ply angle orientation of
the fibers were used as design variables. The complexity of the displacement and strain fields was overcome
through the use of symbolic computation to obtain the corresponding explicit expressions. The sensitivities of
shape design of the element stiffness matrix and load vector were also obtained analytically through symbolic
computation. The model was applied to the structural optimization of multilaminated
axisymmetric shells for
static type situations. Franc0 et al. [58] compared the use of models based on higher-order displacement fields to
first-order and Kirchhoff models on the sensitivity analysis and optimization of laminated plates.
In the present work a family of C” 9-node Lagrangian higher-order discrete models applied to static,
eigenfrequency
and buckling design sensitivity analysis and optimal design of multilaminated
composite plates
is presented. The design variables considered are the ply orientation angles of the fibers and the ply thicknesses.
The design objectives are the minimization
of generalized displacement components, minimization of structure
elastic strain energy, maximization
of natural frequencies of specified vibration modes, maximization
of
buckling load and/or minimization
of structural volume or weight subjected to behavioral constraints. The
sensitivities
of static, eigenfrequency
and buckling response with respect to the design variables can be
evaluated either analytically or by using the semi-analytical
technique [47]. The accuracy and relative computer
efficiency of the developed higher-order models with respect to first-order models and classical plate theory
based models are compared and discussed. Several examples are presented to illustrate the relative performance
of the proposed models.
2. Higher-order
displacement
In order to approximate
displacement components
fields
the three-dimensional
elasticity problem to a two-dimensional
laminate problem, the
U, u and w at any point in the laminate space (Fig. 1) in the X, 4’ and z directions,
Fig. 1. Laminate
geometry
and coordinate
axes
136
C.M.M.
Soores et al.
I Comput. Methods Appl. Mech. Engrg.
149 (1997)
133-152
respectively, are expanded in a Taylor’s series powers of the thickness coordinate
function of X, y, z and t, where t is the time. The following higher-order displacement
z. Each component
fields are considered
is a
[17]
HSDT 11
u=u,+zq+z*u~+z~(P~
v = v, + ze,, + z2v; + z’p;
(1)
w = W” + zq_ + z’wg
HSDT 9
11 =
U” + zq + z2u; + z’$Lq
u = U” + ze, + z*v; + z+
(2)
w = w,,
HSDT 7
u = U” + zq + Z3qq
v = u,, + zq + z$:
(3)
w = WC)
where uO, v,, wO are the displacements
of a generic point on the reference surface, CC),,
fY are the rotations of
normal to the reference surface about the y and x axes, respectively, and (o,, u:, v;S, wg, (p:, (4’1 are the
higher-order terms in the Taylor’s series expansions, defined at the reference surface.
3. Laminate
constitutive
The constitutive
represented by
equations
relations
for an arbitrary
ply k, written
in the laminate
(x, y, z) coordinate
system can be
where qi are the normal and shear stresses in an arbitrary point of kth lamina and E,, and r, are the normal and
shear strains, respectively,
in a arbitrary point in the laminate, referred to the laminate (x, y, z) coordinate
system. The terms of constitutive matrix Qk of ply k are referred to the laminate axes (x, y, z) and can be
obtained from the Q, matrix referred to the fiber directions (l-3) with the transformation
3, = T’Q,T
(5)
where T is the transformation
4. Higher-order
finite element
matrix from the (1,2, 3) fiber axes to the (x, y, z) laminate
axes [59].
formulation
The finite element formulation for the displacement field HSDT 11 represented by Eq. ( 1) applied to 9-node
Lagrangian quadrilateral elements, will be briefly described. The strain components are given by
au
&
xx
=-
ax
au
Yx, =
yielding
&
=YY
au
ay + ax
aV
dy
aW
52
au
Y’z=z+ay
az
aw
au
x7 =
aw
z + yjy
(6)
C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg.
149 (1997)
+
133-152
z3
137
(7)
(8)
A vector @,,, including the bending and membrane terms of the strain components
the transverse shear terms can be defined in such a way as
and a vector ET containing
(9)
(10)
The strains can be written as
[E,_,>
r,, z,s;
11
Yr7
=
(11)
where Z,, and Z, are matrices containing powers of z coordinate (z” with n = 0, . . 3) defined in accordance
with displacement fields and strain relations.
Using C” Lagrangian shape functions [60] the displacements and generalized displacements defined in the
reference surface are obtained within each element, respectively, by
where the Z,
matrix for the displacement
field represented
by Eq. (1) is given by
-100zooz200230
z, =
0
-0
1
0
0
1
0
0
z
0
0
z
0
0
z2
0
0
z2
0
0
z3
01
N, are the Lagrange shape functions of node i and qr is the displacement
displacement vector of the element, qr, by
qe={-.q;.‘.}T,
i=1,...,9
The strains in Eqs. (9) and (10) can be represented
(14)
vector of node i which is related to the
(15)
as
(16)
138
C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg.
149 (1997) 133-152
where B,, and B, are the strain-displacements
matrices, respectively, for bending and membrane
shear, relating the degrees-of-freedom
of the element with the strain components.
The Lagrangian functional for the eth element is
and transverse
(17)
r,;}‘, ~1,= (u, u, w), VI: are the initial stress components, p is the material
where E = {q,., <VY C? Y,~ r,,
density, $ is the work done by the external forces, ti are the generalized velocities and V is the element volume.
Applying Hamilton’s principle and adding the contributions of all finite elements in the domain one obtains
(K + KJq + Mg = F
(18)
where K, M and K, are respectively the structure stiffness matrix, mass matrix and geometric stiffness matrix, q
and 4 are the system displacement and acceleration vectors, respectively. This equation yields for static linear
analysis, free harmonic vibrations and linear elastic buckling analysis, respectively
Kq=F
(19)
Kq-w2Mq=0
(20)
Kq + /U&q = 0
(21)
where o represents the natural frequencies of the structure and A the buckling load parameter. K, M and K,
matrices are obtained by assembling the corresponding element matrices K’, M’ and Kk of the finite elements.
The stiffness and mass matrices of the element are evaluated, respectively, as
(22)
(23)
where 3, is the constitutive matrix in the (x, y, z) laminate axes for kth ply, N is the matrix of the Lagrange
shape functions and pk the material density of kth layer. NL is the number of layers of the laminate, h, is the
vector distance from the middle surface of the laminate to the upper side of kth ply (Fig. 1). Finally, det J is the
determinant of the Jacobian matrix of the transformation
from (5,~) natural coordinates to element (I, y, z)
coordinates.
The element geometric stiffness matrix is given by
(24)
where B, is a strain-displacements
matrix containing the shape functions and their derivatives
matrix containing powers of the z coordinate, both defined in such a way that
and Z,
is a
au au au -----au au au aw aw aw T
--ax'ay'azvax'ay*az'ax? ay' az
ax 1 ay 7 ax ’ ay
The matrix o;, is the initial stress matrix for the kth ply containing all the six stress components
obtained by solving using Eq. (19) and using constitutive relations Eq. (4)
(25)
previously
C.M.M.
Soares et al.
I Comput.
Methods
Appl.
Mech.
Engrg.
149 (1997)
139
I.?.?-IS2
(26)
5. Finite element
models
The finite element model having the displacement field represented by Eq. (I), briefly described above, will
be referred to as Q9-HSDT 11. The remaining elements whose displacement fields are given by Eqs. (2) and
(3) are developed easily from this parent element by deleting the appropriate degrees of freedom leading
respectively, to the finite element discrete models referred to as Q9- HSDT 9 and Q9-HSDT 7. The C” 9-node
Lagrangian first-order discrete model, referred to as Q9- FSDT 5, can also be obtained by deleting all high order
terms and introducing the shear correction factors on the transverse shear terms of constitutive matrix [59].
Two other finite element models are also available in the optimization package. One is the 3-node triangular
shear flexible plate, based on Mindlin’s theory which was developed by Lakshminarayana
et al. [61], usually
referred to as TRIPLT, where the nodal degrees of freedom are the displacements and rotations ug, uO, M”~,,H,, 19,.
and their first derivatives with respect to x and y. The other element is the 3-node triangular discrete Kirchhoff
theory multilaminated
plate-shell element [62], known in the literature as DKT, first developed for isotropic
plates and shells by Batoz et al. [63] and Bathe et al. [64], respectively. The above described finite element
models, corresponding degrees of freedom and number of nodes are shown in Table 1.
6. Sensitivity
analysis
6.1. Statics
For static type situations represented by the equilibrium equation (19) the sensitivities
in the design variables, b, of a generic function sp,= p,(q, b) are evaluated by
with respect to changes
(27)
where 4 is the vector of adjoint displacements
obtained
from
aq
(28)
K4,=ag
The sensitivities
represented
by Eq. (27) can be evaluated
at the element
level by
(29)
Table 1
Finite element models available
for comparison
purposes
Finite element
discrete model
Number of
nodes
Q9-HSDT 11
Q9-HSDT 9
Q9-HSDT 7
Q9-FSDT 5
TRIPLT
9
9
9
9
3
DKT
‘I) The rotation
U”’ VI,. w,,, e,, e,, el”
0: is used only for shell problems.
3
140
C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152
where 4;’ is the adjoint displacements vector for element f, and E the set of elements for which the design
variable bj is defined. The function pi can represent either the objective function of the optimization problem or
a constraint equation. In the first case it can represent the structure elastic strain energy or a generalized
displacement in a specified point. In the second case it can represent a maximum displacement constraint, a
maximum stress constraint or the Hoffman’s failure criterion [59]. A constraint in a generalized displacement
component can be represented by
q = 6i/sman - 1 co
(30)
where C$ is the generalized displacement
component and a,,,,, is the maximum
constraint in a stress component can be represented by
q = +&,
allowed
displacement.
- 1 SO
A
(31)
where 0; is the stress component and cr,,, is the maximum allowed value for that stress component.
Hoffman’s first ply failure criterion, can be represented as a constraint equation by
The
(32)
where u, , o;, Us are the normal stress components in the ( 1, 2,3) fiber axes, u,*, u223,u,, are the shear stress
components, XT, YT, Z, are the lamina normal strengths in tension along the 1,2, 3 directions, respectively, R, S,
T are the shear strengths in the 23, 13 and 12 planes, respectively, and Xc, Yc, Zc are the lamina normal
strengths in compression along the 1,2, 3 directions, respectively. The stress components in the (1,2,3) fiber
axes for the kth ply are obtained from
f”,
v2
g3
(T12
g23
531:
=
T -‘(a,(
2))
=
(33)
T-fek{:;}~e)
The sensitivities of Hoffman’s failure criterion with respect to the design variables are obtained applying
(29). The vector of adjoint forces 4;’ for an arbitrary ply of element 8 is obtained by differentiating
Hoffman’s expression in order to q’ yielding
Eq.
the
The derivatives au, /dq’ are evaluated analytically from the constitutive relations (Eq. (4)). The derivatives
ae / abi are obtained in a similar way, by replacing the terms a?, / dq’ by ks / ab in Eq. (34). The derivatives
au,/ab;are evaluated analytically from constitutive relations.
If the function q represents the structure elastic strain energy
s=lJ=;qTKq
the sensitivities
expression
of the elastic strain energy with respect to design variables
(35)
are obtained
at element level by the
C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152
141
dU
-=
db,
(36)
In the present work the design variables
sensitivities dK’/ ab, are evaluated from
are the ply angles and the ply thicknesses
of the laminate
and the
(37)
where the derivatives
aD/db,
are obtained
either analytically
or by using forward finite difference
1621.
6.2. Free vibrations
The problem of free undamped
the eigenvalue problem
([email protected])@,=0
vibrations
of a finite element discretized
linearly elastic structure is defined by
n=l,...,N
(38)
where N represents the total number of degrees of freedom. The solution of this eigenvalue problem consists of
N eigenvalues of and corresponding eigenvectors O,,, where w,, is the natural frequency of vibration mode n.
For single eigenvalues,
if we consider a vibration mode 0, corresponding
to the natural frequency w,,
normalized through the relation [email protected], = 1, the sensitivity of natural frequency with respect to changes in the
design variable b, is obtained at the element level from
2
aW'
--)o;’
p
(39)
ab,
where 0;’ is the pth mode vector for element 8 and E is the set of elements
defined. The sensitivities HI’ / db, are evaluated from
for which the design variable b, is
(40)
where the derivatives
d(_fi:_, ZT Z Wlab,
are obtained either analytically
or by using forward finite difference.
In the case of multiple eigenvalues they are not, in general, differentiable with respect to design variables and
Eq. (39) becomes inapplicable. This is because the eigenvectors corresponding to the repeated eigenvalues are
not unique. In fact, any linear combination of the eigenvectors will satisfy the eigenvalue problem represented
by Eq. (38). In this case the sensitivities can be obtained by calculating the directional derivatives [65-671 by
considering a simultaneous change of all the design variables.
6.3. Buckling
The buckling
problem
(K+h,K,)O,,==O
of a finite element
n= l,...,
N
discretized
structure
is defined by the eigenvalue
problem
(41)
The solution of this eigenvalue problem consists of N eigenvalues A,, and corresponding eigenvectors O,,, where
A,, is the buckling parameter corresponding to buckling mode n.
For single eigenvalues,
if we consider a buckling mode 0, corresponding
to the buckling parameter AP,
normalized through the relation O%K,Bp = 1, the sensitivity of the buckling parameter with respect to changes
in the design variable b, is obtained at the element level from
(42)
142
C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg 149 (1997) 133-152
The sensitivities
dK’,/ ab, are evaluated
by
~=ri~~~~~~,~(~~;~Z~[~
i ~-&&)]J&detJdidv
(43)
where the derivatives da;,/abi are evaluated by Eqs. (28)-(29).
For multiple eigenvalues
the sensitivities must be obtained by calculating
7. Optimal
the directional
derivatives
]65-671.
design
The structural
optimization
problem
can be stated as
min{ LI(b )}
subject to:
bi -C b, C by
i = 1, . . , ndu
$.(q,b)SO
j=
l,...)
m
(44)
where L?(b) is the objective function, +,(q, b) are the m inequality constraint equations, bj and by are,
respectively, the lower and upper limits of the design variables and ndv is the total number of design variables.
The objective function can be the structural weight or volume, the generalized displacement in a given node, the
elastic strain energy of the structure, the natural frequency of a given vibration mode or the buckling load
parameter.
The constrained optimization problems are solved by using the method of feasible directions [68] and the
unconstrained
problems are solved by using the BFGS (Broydon-Fletcher-Goldfarb-Shanno)
method 1681.
Ply angles and ply thicknesses can also be designed using a two level procedure [62,69,70] schematically
min R
subject to:
i = I,...,ndv
bi’ < bi I b;
Design variables : Ply angles
I
min Q
subject to:
bf S bi I b/’
i = l,...,ndv
j = l,...,m
w~G-IM~ 0
Design variables : Ply thicknesses
Fig. 2. Two-level
optimization
process.
C.M.M.
Soares et al. I Comput. Methods Appl. Mech. Engrg.
149 (1997)
II-IS2
143
described in Fig. 2. In this case the global objective is to minimize the weight/volume
of the structure. At the
first level of optimization, the weight /volume of the structure is kept constant and the optimal fiber directions
are determined for minimum deflection, minimum elastic strain energy, maximum natural frequencies of given
modes or maximum buckling load. At the second level the optimal ply thicknesses are found assuming the
previously obtained ply angles. In this second level the weight/volume
of the structure is minimized subject to
structural behavior constraints as well as lower and upper bounds on design variables.
8. Numerical
8.1. Dejection
applications
and stresses in simply supported
laminated plates
The prediction of deflections and stresses of the present higher-order discrete models is compared with exact
solutions obtained by Pagan0 and Hatfield [71] for a 3-Ply [0”/90”/0”]
symmetric square (a X a) simply
supported plate subjected to sinusoidal surface load q(x, y) given by
q(x,y)=q,,sin($)sin(~),
OSxCa,
OGyCa,
-h/2szch/2
The thickness of 1st and 3rd plies is h/4 each and the thickness of 2nd ply is h/2, where h is the total thickness
of the laminate. The following material properties are used: E, = 25 X 10h psi, E, = E, = 10h psi, G,? = G, 3 =
0.5 X lo6 psi, G,, = 0.2 X 10h psi, v,~ = v,~ = ~23 = 0.25.
The nondimensional
maximum deflections of the 3-ply plate, for several length-to-thickness
ratios (u/h),
obtained with the present higher-order models are compared in Table 2 with the exact solutions of Pagan0 and
Hatfield [71]. Alternative solutions obtained with discrete models based on first-order theory and discrete
Kirchhoff theory (DKT) are also presented. A quarter plate with a 4x4 finite element mesh was used for all
models. Percentage errors relative to Pagan0 and Hatfield results are shown, ranging from 1.3 to 4.4% in the
case of higher-order
models and achieving 7% and 10.6% in the case of first-order models for lower
length-to-thickness
ratios (a/h = 4). In the case of discrete Kirchhoff theory model it can be seen that the
element is not applicable for length-to-thickness
ratios lower than 50.
The nondimensional
stresses obtained with the present models are compared in Table 3 with the exact results
from Pagan0 and Hatfield [71]. A good agreement was found for higher-order models even for very low a/h
ratios where first-order models presented errors up to 50% for example in the prediction of normal stress v,,.
This error decreases as the a/h ratio increases. The higher-order models have shown good results on shear
transverse stresses for low a/h ratios, but for a/h ratios higher than 50 lower accuracy is achieved.
Table 2
Maximum
nondimensional
deflections
of 3.~1~ simply supported
p = 4G,, + (E, + E2( 1 + 2u,,))/
square plates w.* = ~‘pl”l(l2(alh)“hy,,),
(1 - VII1l?,)
nlh
4
Pagano et al. [71]
Q9-HSDT II
Q9-HSDT
9
Q9-HSDT
7
Q9-FSDT
5
TRIPLT
DKT
Percentage
IO
4.49 I
4.2894
(-4.4%)
1.709
I .6437
(-3.8%)
4.3545
(-3%)
4.3545
(-3%)
I .6492
(3.5%)
I .6492
(3.5%)
4.1739
(-7%)
4.0139
(- 10.6%)
0.9759
(-78%)
I s975
(-6.5%)
I s403
(-9.9%)
0.9759
(-43%)
errors with respect to Pagan0 et al. [7l] solution.
20
SO
100
I. I 89
I .03I
I.1615
(-2.3%)
I.0157
(- 1.5%)
1.1626
(-2.2%)
I.1626
(-2.2%)
1.1477
(-3.5%)
I.1080
(-6.8%)
0.9759
(-18%)
I.0159
(- 1.4%)
1.0159
(- I .4%)
I.0134
(- 1.8%)
0.9789
(-5%)
0.9759
(-5.3%)
I .OQ8
0.9944
(- 1.3%)
0.9945
(- 1.3%)
0.9945
(- 1.3%)
0.9939
(- 1.4%)
0.9596
(-4.8%)
0.9759
(-3.1%)
144
C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-1.52
Table 3
Nondimensional
stresses of 3-ply simply supported
ex
alh
4
square plates (cr:otcrX)
u:
(a/2,a/2,
+h/2)
= 1/(4,(a/h)‘)(~~~“~~“),
u:,
(a/2,a/2,
*h/4)
*
(~*;a:)
=
14q,,Wh))(q,;~,J
*
(O,O, *h/2)
;0;;2,0,0)
fd,o/2,0)
0.720, -0.684
0.725, -0.688
0.706, -0.706
0.706, -0.706
0.371/k0.371
0.352, -0.352
0.663, -0.666
0.616/-0.619
0.631/-0.631
0.6311-0.631
0.656, -0.656
0.625 / -0.625
-0.0467/0.0458
-0.0455/0.0446
-0.0461/0.0461
-0.0461/0.0461
-0.0333/0.0333
-0.0319/0.0319
0.292
0.249
0.249
0.249
0.194
0.161
0.219
0.212
0.214
0.214
0.155
0.135
Parano et
Q9-HSDT
Q9-HSDT
Q9-HSDT
Q9-FSDT
TRIPLT
al. [71]
11
9
7
5
10
0.559,
0.560,
0.562,
0.562,
0.488,
0.464,
-0.559
-0.561
-0.562
-0.562
-0.488
-0.464
0.401, -0.403
0.389/-0.391
0.391/-0.391
0.3911-0.391
0.387, -0.387
0.367/-0.367
-0.0275lO.0276
-0.0272/0.0271
-0.027310.0273
-0.0273iO.0273
-0.0249/0.0249
-0.0237/0.0237
0.196
0.171
0.170
0.170
0.130
0.093
0.301
0.286
0.286
0.286
0.197
0.175
Pagan0 et
Q9-HSDT
Q9-HSDT
Q9-HSDT
Q9-FSDT
TRIPLT
al. [71]
11
9
7
5
20
0.543,
0.545,
0.545,
0.545,
0.525 /
0.497,
-0.543
-0.543
-0.545
-0.545
-0.525
-0.497
0.308/-0.309
0.305/-0.305
0.305 / -0.305
0.305/-0.305
0.303/-0.303
0.288, -0.288
-0.0230/0.0230
-0.0229lO.0229
-0.0229/0.0229
-0.0229/0.0229
-0.0223/0.0223
-0.0212/0.0212
0.156
0.164
0.164
0.164
0.133
0.077
0.328
0.318
0.318
0.318
0.219
0.195
Pagan0 et al. [71]
Q9-HSDT 11
Q9-HSDT 9
Q9-HSDT 7
Q9-FSDT 5
TRIPLT
50
0.539/-0.539
0.540, -0.540
0.540, -0.540
0.540, -0.540
0.536, -0.536
0.5061-0.506
0.276,
0.276,
0.276,
0.276,
0.275,
0.258,
-0.0216/0.0216
-0.0215/0.0215
-0.0215/0.0215
-0.021510.0215
-0.0214/0.0214
-0.0202/0.0202
0.141
0.324
0.324
0.324
0.298
0.105
0.337
0.388
0.388
0.388
0.288
0.211
Pagan0 et al. [71]
Q9-HSDT 11
Q9-HSDT 9
Q9-HSDT7
Q9-FSDTS
TRIPLT
-0.276
-0.276
-0.276
-0.276
-0.275
-0.258
8.2. Free vibration of simply supported
square laminated plates
In order to compare the accuracy on the prediction of natural frequencies of higher-order models (HSDT) and
first-order models (FSDT) a few test cases are presented emphasizing the effect of the degree of orthotropy of
individual layers (Young’s modules ratio E, /E,) and the effect of side-to-thickness
ratios. The results obtained
are compared with alternative solutions when available.
To analyze the effect of the degree of orthotropy of individual layers we consider the free vibration problem
of a simply supported laminated square plate having the following material properties: E, /E, = 3, 10, 20, 30
and 40; E, = E, = 10 GPa; G,,/E,
= G,, /E, = 0.6; G,, /E, = 0.5; v,~ = v,~ = y3 = 0.25. The side-to-thickness ratio is a/h = 5. The following
stacking sequences are considered:
4-ply [0”/90”/0”/90”],
6-ply
[O”/9O”/O”/9O”/O”/9O”]
and lo-ply [O”/9~/Oo/900/oO/900/O”/9~/Oo/90”].
A quarter plate 4 X 4 finite
element mesh is used.
Table 4 compares the results for the nondimensional
fundamental natural frequency W obtained with HSDT
models, FSDT models, discrete Kirchhoff model [62] (DKT) and a 3-D elasticity solution obtained by Noor
[72]. It can be seen that the results obtained with DKT model are not acceptable with errors increasing for
higher E, /E, ratios. The errors obtained with FSDT models increase for higher E, /E2 ratios achieving errors of
about 5.8%.
In Table 5 are presented the nondimensional
natural frequencies corresponding
to the symmetric modes
higher than the fundamental one for the antisymmetric 4-ply [O”/90”/O”/90”]
and also for a symmetric 5-ply
[0”/90”/0” /90”/0’]
lamination sequence assuming an E, /E2 ratio equal to 40 and a/h = 5. Discrepancies of
FSDT models with respect to higher-order model Q9-HSDT 11 are presented. It can be observed that FSDT
models accuracy on the prediction of natural frequencies decrease when higher vibration modes are considered
achieving discrepancies of about 10% for the second vibration mode.
C.M.M. Soar-es et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-1.52
Table 4
Nondimensional
NL
4
fundamental
Noor [72]
Q9-HSDT I I
Q9-HSDT 9
7
Q9-FSDT 5
TRIPLT
DKT
6
Noor [72]
Q9-HSDT I I
QV-HSDT 9
Q9-HSDT 7
Q9-FSDT 5
TRIPLT
DKT
IO
Noor [72]
Q9-HSDT I I
Q9-HSDT
9
Q9-HSDT
7
Q9-FSDT
5
TRIPLT
DKT
Percentage
(w = w=
Degree of orthotropy
Model
Q9-HSDT
natural frequencies
X IO) of symmetric
of individual
simply supported
145
square plates (a/h = 5)
layers E, lE2
3
IO
2.6182
2.6059
(-0.47%)
2.5983
(-0.76%)
2.6004
(-0.68%)
2.6018
(-0.63%)
2.6047
(-0.52%)
3.0174
(15.2%)
3.2578
3.2594
(0.05%)
3.2513
(-0.20%)
3.2780
(0.62%)
3.2899
(0.99%)
3.2999
(1.29%)
3.7622
3.7871
(0.66%)
3.7793
(0.46%)
3.8502
(2.34%)
3.87.55
(3.01%)
3.8873
(3.33%)
5.2975
(40.8%)
4.0660
4.1094
( I .07%)
4.1022
(0.89%)
4.2135
(3.63%)
4.2480
(4.48% j
4.2598
(4.77%)
4.2719
4.3289
( I .33%)
4.3224
(1.18%)
4.4683
(4.60%)
4.5084
(5.54%)
4.5 I98
(5.80%)
7.0879
(65.9%)
2.6440
2.6287
(-0.58%)
2.621 I
(-0.87%)
2.6224
(-0.82%)
2.6229
(-0.80%)
2.6240
(-0.76%)
3.0466
(152%)
3.3657
3.3546
(-0.33%)
3.3467
(-0.57%)
3.3622
(-0.11%)
3.3674
(0.05%)
3.3726
(0.21%)
3.9359
3.9342
(-0.04%)
3.9267
(-0.23%)
3.9673
(0.80%)
3.9772
( I .05%)
3.9841
( I .22%)
55364
(40.7%)
4.2783
4.2848
(0.15%)
4.2782
(-0.002%)
4.3420
( I .49%x)
4.3532
(I .75%)
4.3607
( I .93%)
4.509 I
4.5223
(0.29%)
45164
(0.16%)
4.6005
(2.03%)
4.6 I06
(2.25%)
4.6184
(2.42%)
7.46SS
(65.6%)
2.6583
2.6409
(-0.66%)
2.6332
(-0.94%)
2.6338
(-0.92%)
2.6336
(-0.93%)
2.6337
(-0.92%)
3.0615
(15.2%)
3.4250
3.4066
(-0.54%)
3.3988
-0.76%)
3.405 I
-0.58%)
3.4054
-0.57%)
3.4082
-0.49%)
4.0337
4.0 I76
(-0.40%)
4.0105
(0.57%)
4.0269
(-0.17%)
4.0256
(-0.20%)
4.0301
(-0.09%)
5.6556
(40.2%)
4.401 I
4.3880
(-0.30%)
4.3818
(-0.44%)
4.4075
(0.15%)
4.4024
(0.03%)
4.4078
(0.15%)
4.6498
4.6398
(-0.22%)
4.6344
(-0.33%)
4.6683
(0.40%)
4.6578
(0.17%)
4.6640
(0.30%)
7.6528
(64.6%)
20
30
40
errors with respect to Noor (721 solution.
8.3. Buckling
of simply supported
laminated plates
The accuracy on the prediction of buckling loads of higher-order models (HSDT) and first-order models
(FSDT) is analyzed and compared with alternative solutions for a simply supported square plate subjected to
uniaxial membrane uniform compressive load lV1. The effect of the degree of orthotropy E, /E, is investigated.
The following material properties are considered: E, /E, = 3, 10, 20, 30 and 40; E, = E, = 10 GPa; G,*/E2 =
G,,fE,
=0.6; G,,/E,
=0.5; q2 = u,~ = vzj = 0.25. The side-to-thickness
ratio is kept constant a/h = 10. A
full plate 4 X 4 finite element mesh is used. The nondimensional
buckling loads A for an anti-symmetric
4-ply
[0”/90”/0”/90”]
and for a symmetric S-ply [0”/90”/0”/90”/0”]
are shown in Tables 6 and 7, respectively. The
solutions obtained with HSDT and FSDT models are compared to a 3-D elasticity solution [36], a local
higher-order deformation theory obtained by Wu and Chen [36] and a classical plate theory (CPT) [36]. The
146
C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152
Table 5
Nondimensional
E, IE, = 40
NL
natural
frequencies
Model
(w = wr ph /E2 X IO) of a simply
Vibration
Q9-HSDT I I
Q9-HSDT 9
Q9-HSDT 7
Q9-FSDT 5
TRIPLT
Q9-HSDT 1 I
Q9-HSDT 9
Q9-HSDT 7
Q9-FSDT 5
5
TRIPLT
Percentage
Table 6
Nondimensional
uniaxial
Model
buckling
2
12.0385
12.0559
12.8706
12.6612
(52%)
12.7586
(6.0%)
12.0385
12.0.559
12.8706
12.7115
(5.6%)
12.7.586
(6.0%)
4.5517
4.5s 19
4.55 19
4.5847
(0.6%)
45896
(0.7%)
1 I .5439
I I.5214
12.3709
12.6486
(9.6%)
12.7357
( 10.3%)
13.4340
13.4316
13.Sll7
12.71 I5
(-54%)
12.8009
(-4.7%)
11
Q9-HSDT
9
Q9-HSDT
7
Q9-FSDT 5
3-D elasticity [36]
Wu and Chen [36]
CPT [36]
Percentage
Table 7
Nondimensional
load A = N\u’l(f+‘)
buckling
1I
Q9-HSDT
9
Q9-HSDT
7
Q9-FSDT 5
3-D elasticity [36]
Wu and Chen [36]
CPT [36]
Percentage
for an anti-symmetric
of individual
5
16.5854
16.5516
17.6423
17.3009
(4.3%)
17.5745
(6.0%)
20.7687
20.904 I
22.3941
2 I .3789
(2.9%)
2 I .6656
(4.3%)
20.7687
20.904 I
22.3941
21.3789
(2.9%)
21.6656
(4.3%)
17.1171
17.1091
17.7173
17.31 IO
(1.1%)
17.5770
(2.7%)
19.9366
19.9179
21.4123
2 1.2922
(6.8%)
21.5693
(8.2%)
22.9358
22.9376
23.0846
2 I .4788
(-6.4%)
21.7609
(-5.1%)
6
4.~1~ [0”/90”/0”/90”]
simply supported
square plate
layers, E, lEZ
10
20
30
40
5.1356
(-0.74%)
5.1308
(-0.83%)
5.1527
(-0.41%)
5.1575
(-0.32%)
5.1738
5. I739
5.5738
(7.7%)
9.0263
(0.1 I%)
9.0222
(0.06%)
9.1216
(1.2%)
9.1769
(1.8%)
9.0164
9.0176
10.2947
( 14.2%)
13.7838
(0.29%)
13.7804
(0.27%)
14.0708
(2.4%)
14.2407
(3.6%)
13.7429
13.7461
16.9882
(23.6%)
17.8528
(0.39%)
17.8523
(0.39%)
18.3972
(3.5%)
18.7088
(5.2%)
17.7829
17.7886
23.6746
(33.1%)
21.3756
(0.45%)
2 I .3799
(0.47%)
22.22 14
(4.4%)
22.6889
(6.6%)
2 I .2796
2 I .2880
30.359 I
(42.7%)
solution.
load A = N<a’l(Ezh’) for a symmetric
Degree of orthotropy
Q9-HSDT
4
3
errors with respect to 3-D elasticity
Model
ratio a/h = 5 and
11 model.
Degree of orthotropy
QB-HSDT
3
4.3289
4.3224
4.4683
4.5084
(4.1%)
45198
(4.4%)
errors with respect to Q9-HSDT
square plate with side-to-thickness
mode
I
4
supported
5-ply [0”/90”/0”/90”/0”]
of individual
stmply supported
square plate
layers, E, lE2
3
10
20
30
40
5.2816
(-0.82%)
5.2709
(-1.0%)
5.2807
9.9424
(-0.18%)
9.9230
(-0.37%)
9.9509
(-0.09%)
9.9417
(-0.18%)
9.9603
9.985 I
11.4918
(15.4%)
15.6716
(0.12%)
15.6418
(-0.07%)
15.7016
(0.31%)
15.6787
(0.12%)
15.6527
15.6934
19.7124
(25.9%)
20.5375
(0.35%)
20.5013
(0.17%)
20.5954
(0.63%)
20.5705
(0.51%)
20.4663
20.5 I76
27.9357
(36.5%)
24.7257
(0.54%)
24.6863
(0.38%)
24.8151
(0.9%)
24.8049
(0.86%)
24.5929
24.65 I7
36.1597
(47.0%)
(-0.84%)
5.2801
(0.03%)
5.3255
5.3323
5.7538
(8.0%)
errors with respect to 3-D elasticity
solution
C.M.M.
Soares et al.
I Compui.
Methods
Appl.
Mech.
Engrg.
149 (1997)
133-152
147
percentage errors with respect to 3-D elasticity solution are shown. The errors obtained with CPT solution
increase for higher E, /E, ratios achieving 42.7%. The errors presented by FSDT model can be about 6% higher
than the higher-order models Q9-HSDT 11 and Q9-HSDT 9 in the anti-symmetric
case. In the symmetric case
the results obtained with HSDT and FSDT models are both very good.
8.4. Optimal ply angles ,for maximum
buckling
load of rectangular
plates
A symmetric simply supported rectangular plate (a X 6) made of 4 plies of equal thickness with the
lamination sequence [H/-0/-0/0]
and having the following material properties: E, = 142.5 GPa, Ez = E, =
= 0.25 and the total thickness h = 0.04 m is
9.79GPa, GIZ=G,,=4.72GPa,
G,,= l.l92GPa,
v,~=v,~=I/?~
now designed for maximum uniaxial buckling load N,. A linking relation between ply angles is imposed in order
to have only one design variable. The angle 8 is measured counterclockwise
from the x axis. Table 8 shows the
optimal ply angle tl for maximum uniaxial buckling load, for several length-to-width
ratios b/a and the
corresponding
value of the buckling load. A good agreement is found between all higher-order models. The
t&t-order model shows some discrepancies lower than 2%.
8.5. Optimal design of square plate with central circular hole
It is considered the optimal design of a simply supported laminated square plate with a central circular hole
using a two level optimization process. The laminate is made up of 6 plies of equal thickness t, = 0.015 m and is
subjected to an uniform pressure p, = 50000 N/m’. The side dimension of the plate is a = 2 m and the hole
diameter is a/3. A full finite element model with 288 nodes and 64 elements is used (Fig. 3).
At the first level of optimization the objective function is the minimization of the plate elastic strain energy
and the design variables are the ply angles. The weight of the structure is kept constant. To accomplish this
Table 8
Optimal ply angle B for maximum
hlc1
I
1.2
I .3
I .‘l
I .s
I .6
1.7
I .8
2
Q9-HSDT
uniaxial
buckling
load A’$of simply supported
plates (u/h = 25)
Q9-HSDT 7
Q9-HSDT 9
II
rectangular
Q9-FSDT 5
Ply
angles
N>
(kN)
PlY
angles
N!
(kN)
PlY
angles
N\
(kN)
Ply
angles
NI
(kN)
30.9”
45.5”
43.9”
42.3”
40.7”
39.4”
38.4”
38.2”
39.9”
9944.94
9745.04
9820.03
9863.92
9880.08
9872.3 I
9845.21
9806.64
9750.27
30.9”
45.5”
43.9”
42.2”
40.7”
39.4”
38.4”
38.2”
39.9”
9945.04
9735.06
98 IO.46
9854.65
9870.93
9863.17
9836.09
9797.64
9741.33
30.8”
45.1”
43.6
42.0”
40.5”
39.1”
38.1”
37.7”
39.3”
10027.9 I
9762.23
9841.18
9889.85
9910.61
9906.79
9882.49
9844.60
3 I .5”
45.5”
44.0
42.4”
40.9”
10194.58
9954.97
10030.57
10074.30
10089.24
10079.43
10050.04
10009.49
9949.84
9782.15
39.6
38.7”
38.5”
40.1”
(b)
Fig. 3. Square plate with central circular
hole: (a) finite element mesh; (b) plate divided into 16 regions.
C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152
148
unconstrained optimization problem the plate is divided into 16 regions leading to 96 design variables (Fig. 3).
The ply angles for all elements lying in one region are equal. In the initial design the ply angles are all set to 0“
(fibers aligned with x axis).
At the second level of optimization the objective function is the minimization of the plate volume subject to
Hoffman’s failure criterion with a stress safety factor of 2.5 and maximum deflection constraints with
S,,, = 0.005 m. The design variables are the ply thicknesses. The thickness of each ply is assumed to be
constant over the plate domain and the ply angles are kept constant at this optimization stage. The material
properties are representative of those of a High-modulus Graphite/Epoxy
with a fiber volume fraction of 0.6:
E, = 220 GPa, E, = E, = 6.9 GPa, G,, = G,, = G,, = 4.8 GPa, v,~ = v,, = v,? = 0.25, p = 1640 Kg/m’. The
strength properties are: X, = 760 MPa, YT= Z, = 28 MPa, Xc = 690 MPa, Yc = Z, = 170 MPa, R = S = T =
70 MPa. The upper and lower limits on the ply thicknesses are, respectively, 0.1 m and 0.001 mm.
The optimal ply angles and ply thicknesses obtained with higher-order models and first-order model are
presented in Table 9. Due to the symmetry of the optimal ply angles with respect to x and y axes, only the
results for regions located on 1st quadrant are shown. A good agreement between all HSDT models is found and
some discrepancies in the thickness distribution are obtained with the QPFSDT 5 model. Table 9 shows also the
average CPU time ratios of HSDT models with respect to FSDT model.
8.6. Optimal
design of cantilever
panel
The cantilever panel represented in Fig. 4 is designed for minimum volume subject to Hoffman’s stress failure
criterion. The laminated panel is made up of 5 plies having the lamination sequence [ -20”25”/70”/25”/
-2O”].
The panel is subjected to an uniform pressure p; = 10 000 N/m*. The material properties are E, = 290 GPa,
E,=E,=6.2
GPa, G,,=G,,=G23=4.8
GPa, u,,= y1 = v,, = 0.25, p = 1700 Kg/m’. The strength properties are: X, = 620 MPa, YT = Z, = 2 1 MPa, X, = 620 MPa, Yc = Z, = 170 MPa, R = S = T = 60 MPa.
The objective is to minimize the volume of panel and find the optimal average thickness distribution subject
to Hoffman’s first ply failure criterion. A strength safety factor of 2 is used. The panel is divided into 8 regions
as shown in Fig. 4 and the thicknesses are constant inside each region. Thus, the optimization problem has
5 X 8 = 40 design variables. In the initial design all thicknesses in each ply are set to 12 mm. The upper and
lower limits on the ply thicknesses are, respectively, 0.1 m and 0.001 mm. The optimal thickness distribution for
regions 1, 2, 3 and 8 obtained with HSDT and FSDT models is presented in Table 10. It can be observed that a
close agreement is obtained between all HSDT models but discrepancies of about 41% in the final volume are
obtained with Q9-FSDT 5 model. In this example the lower accuracy in the stress prediction of FSDT model
leads to an unsafe design. The average CPU time ratios with respect to FSDT model are also presented in Table
10.
Table 9
Two-level
optimization
results for square plate with central circular
hole
Model
Q9-HSDT
Optimal ply angles
for 1st quadrant
[ -6.7”/ -0.9”],
[-25.0”/-3.5”].
[-33.2”/ -4.7”]\
[-9.3”/-1.4”1\
[-6.7”/-0.9”]\
[-25.0”/-3.5”]\
[-33.2”/-4.7”]\
[-9.3”/-1.4”],
[ -6.7”/ -0.9”]\
[ -6.6”/ -0.9”]\
[-25.0”/-3.5”]\
[-33.2”1-4.7”]\
[-9.3”/1.4”]>
[-25.2”/-3.6”],
[-33.7”/-4.8”]\
[-9.5”/-1.4”]\
Optimal thicknesses
r,, t2,t,, I, (4
0.01435
0.00875
0.00967
0.01404
0.01512
0.00829
0.00829
0.01512
0.01512
0.00829
0.00829
0.01512
0.01193
0.01 153
0.01 153
0.01193
Initial volume (m’)
Final volume (m’)
0.21906
0.17091
0.21906
0.17096
0.2 I906
0.17096
0.21906
0.17131
CPU time ratio
7.8
3.8
2.0
I
II
Q9-HSDT
9
Q9-HSDT
7
Q9-FSDT 5
149
C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) II-152
a
Clamped
2
3
4
(a)
Fig.
Table
panel:
(a) finite
element
mesh
and dimensions
(m);
(b) panel
divided
into 8 regions
IO
Average
thickness
Region
distribution
for minimum
Ply
I
8
Initial
4. Cantilever
volume
(m’)
Final
volume
(m‘)
CPU
time ratio
volume
of a cantilever
Average
thickness
Q9-HSDT
II
panel
subiected
distribution
to Hoffman’s
stress failure
criterion
(m)
Q9-HSDT
9
Q9-HSDT
7
Q9-FSDT
I
0.009766
0.009766
0.009766
0.008683
2
0.009754
0.009755
0.009754
0.00847
3
0.009756
0.009756
0.009754
0.008477
4
0.009758
0.009758
0.009754
0.008430
5
0.009769
0.009769
0.009766
0.008437
I
0.009466
0.009467
0.009466
0.007569
2
0.009455
0.009456
0.009454
0.007268
3
0.009456
0.009457
0.009454
0.007282
4
0.009457
0.009457
0.009454
0.007237
5
0.009468
0.009468
0.009466
0.007344
5
I
I
0.009208
0.009208
0.009203
0.006 I4 I
2
0.009206
0.009206
0.009
0.006044
3
0.009201
0.009201
0.009199
0.006057
4
0.009195
0.009
0.009199
0.006044
5
0.009201
0.00920
0.009203
0.006133
I
0.00823
2
0.008256
3
0.0082
4
5
I
I95
I
I99
0.008230
0.008202
0.008255
0.0082
0.008217
0.00822
0.008181
0.008181
0.008225
0.001932
0.008173
0.008
0.008208
0.001949
0.360
0.360
0.360
0.360
0.270544
0.27053
0.270544
0.158477
5.6
3
I.8
I
I8
172
I
0.00 1950
I9
I
0.001931
0.002278
150
C.M.M.
Soares et al.
I Comput.
Methods
Appl. Mech.
Engrg.
149 (1997)
133-152
9. Conclusions
A family of C” Lagrangian finite element models based on a refined shear deformation theory assuming a
nonlinear variation for the displacement field has been developed. These models have been incorporated in an
optimization package in order to obtain the structural sensitivities of response with respect to changes in design
variables (ply angles and ply thicknesses) and to carry out the structural optimization of multilaminated
plates,
considering static, free vibration and buckling constraints and/or objective functions. The optimization process
can be developed by using a two level scheme.
Numerical illustrative applications have shown that higher-order models are able to accurately predict the
behavior of highly anisotropic and/or low length-to-thickness
ratio laminated plates with reasonable advantage
over first-order models. Results presented in Tables 2 and 4 show that Kirchhoff models are unable to predict
the behavior of highly anisotropic and/or low length-to-thickness
ratio laminated plates.
The quadrangular
9-node Lagrangian elements with higher-order displacement
fields have shown good
accuracy but the computational
efficiency decreases when more complex displacement
fields are used.
First-order models have shown poor accuracy on stress prediction. Global constraints and/or objectives such as
natural frequencies of specified vibration modes and buckling loads obtained with HSDT models are in better
agreement with available 3-D elasticity solutions than the FSDT models (Q9-FSDT 5 and TRIPLT). Natural
frequencies of free vibration modes higher than the fundamental one are better predicted by using the present
refined HSDT models. The higher-order model with nine degrees-of-freedom
per node Q9-HSDT 9 seems to
represent a reasonable compromise between accuracy and computational efficiency.
The analytical sensitivities in order to ply angles and ply thicknesses are easily and efficiently obtained for
discrete models based on higher-order displacement fields. The use of FSDT models in optimization problems
with first ply failure constraints can lead to unsafe designs. Higher-order theories are an improvement over the
first-order theory in order to accurately predict the behavior of laminated plates with low length-to-thickness
ratios and/or high degree of anisotropy. The use of higher-order discrete models in the optimal design of
multilayered composite plates and sandwich plates is very promising as these models can be easily implemented
in existing structural optimization packages.
Acknowledgments
Sponsorship from the following grants is gratefully acknowledged:
Human Capital and Mobility Project
‘Diagnostic and Reliability of Composite Materials and Structures for Advanced Transportation
Applications’
(Project CHRTX-CT93-0222)
and Fundacao Luso-Americana
para o Desenvolvimento
(FLAD).
References
[I] R.D. Mindlin, Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. J. Appl. Mech., Trans. ASME 18
(1951) 31-38.
[2] J.N. Reddy, A review of refined theories of laminated composite plates, Shock Vib. Dig. 22(7) (1990) 3-17.
[3] J.M. Whitney, Shear correction factors for orthotropic laminates under static load, J. Appl. Mech. 40( 1) (1973) 302-304.
[4] K.H. Lo, R.M. Christensen and E.M. Wu, A high-order theory of plate deformation. Part I: Homogeneous plates, J. Appl. Mech.
(1977) 663-668.
[5] K.H. Lo, R.M. Christensen and E.M. Wu, A high-order theory of plate deformation. Part 2: Laminated plates, J. Appl. Mech. ( 1977)
669-676.
[6] J.N. Reddy, A simple higher-order theory for laminated composite plates, J. Appl. Mech. 51 (1984) 745-7.52.
[7] J.N. Reddy, On refined computational
models of composite laminates, Int. J. Numer. Methods Engrg. 27 (1989) 361-382.
[8] J.N. Reddy, An evaluation of equivalent single-layer and layerwise theories of composite laminates, Composite Struct. 2.5 (1993)
31-35.
[9] N.D. Phan and J.N. Reddy, Analysis of laminated composite plates using a higher-order shear deformation theory, Int. J. Numer.
Methods Engrg. 21 (1985) 2201-2219.
[IO] L. Librescu, Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures (Noordhoff, Leyden, Netherlands,
1975).
[ 1 I] R. Schmidt, A refined non-linear theory of plates with transverse shear deformation, J. Ind. Math. Sot. 27(l) (1977) 23-38.
C.M.M. Soares et al. I Comput. Methods Appl. Mech. Engrg. 149 (1997) 133-152
I51
[ 121 A.V. Krishna Murty, Higher order theory for vibration of thick plates, AIAA J. 1% 12) (1977) 1823-1824.
[I.?] M. Levinson, An accurate. simple theory of the statics and dynamics of elastic plates, Mech. Res. Comm. 7(6) (1980) 343-350
[ 141 P. Seide, An improved approximate theory for the bending of laminated plates, Mech. Today 5 ( 1980) 451-466.
[IS] M.VV Murthy, An improved transverse shear deformation theory for laminated anisotropic plates, NASA Tech. Paper 1903 ( 19811
I-37.
[ 16) A. Bhimaraddi
[ 171
[ 181
[I91
[20]
[2l]
[22]
[23]
1241
[25]
[26]
[27]
[28]
[29]
[30]
[3l]
[32]
[33]
[34]
[35]
[36]
and L.K. Stevens,
A higher-order
theory for free vibration
of orthotropic
homogeneous
and laminated
rectangular
plates, J. Appl. Mech. 51 (1984) 195-198.
Mallikarjuna and T. Kant, A critical review and some results of recently developed relined theories of fiber-reinforced
laminated
composites and sandwiches, Composite Struct. 23 (1993) 293-312.
T. Kant and B.N. Pandya, A simple finite element formulation of a higher-order theory for unsymmetric laminated composite plates,
Composite Structur. 9 (1988) 215-224.
J.N. Reddy, On refined theories of composite laminates, Meccanica 25(4) (1980) 230-238.
A.K. Noor and W.S. Burton, Assessment of computational
models for multilayered anisotropic plates, Composite Struct. I4 (1990)
233-265.
C.W. Bert, A critical evaluation of new plate theories applied to laminated composites, Composite Struct. 2 (1984) 329-347.
T. Kant and J.R. Kommineni, Geometrically
nonlinear analysis of symmetrically
laminated composite and sandwich shells with
higher-order theory and C” finite elements, Composite Structures 27 (1984) 403-418.
J.N. Reddy and D.H. Robbins Jr., Theories and computational
models for composite laminates, Appl. Mech. Rev. 47(6) (1994)
137-169.
D.H. Robbins Jr. and J.N. Reddy, Variable kinematic modeling of laminate composite plates, hit. J. Numer. Methods in Engrg. 39
(1996) 2283-2317.
J.N. Reddy and N.D. Phan, Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear
deformation theory, J. Sound Vib. 98(2) (1985) 157-170.
J.N. Reddy and A.A. Khdeir, Buckling and vibration of laminated composite plates using various plate theories. AIAA J. 2( 12) ( 1989)
1808-1817.
A.A. Khdeir, Free vibration and buckling of unsymmetric cross-ply laminated plates using a refined theory, J. Sound Vib. 128(3)
(1989) 377-395.
N.R. Senthilnathan,
S.P. Lim, K.H. Lee and S.T. Chow, Vibration of laminated orthotropic plates using a simplilied higher-order
deformation theory, Composite Struct. 10 (1988) 211-229.
T. Kant, RV. Ravichandran, B.N. Pandya and Mallikarjuna, Finite element transient dynamic analysis of isotropic and tibre reinforced
composite plates using higher-order theory, Composite Struct. 9 (1988) 319-342.
N.S. Putcha and J.N. Reddy, Stability and natural vibration analysis of laminated plates by using a mixed element based on a refined
plate theory, J. Sound and Vib. 104 (1986) 285-300.
F. Kozma and 0.0. Ochoa, Buckling of composite plates using shear deformable finite elements, AIAA J. 24( IO) (1986) 1721-1723.
S. Liu, A vibration analysis of composite plates, Finite Elem. Anal. Des. 9 (1991) 295-307.
Mallikarjuna and T. Kant, Free vibration of symmetrically laminated plates using a higher-order theory wnh tinite element technique,
Int. J. Numer. Methods in Engrg. 28 (1989) 1875-1889.
A.K. Ghosh and S.S. Dey, A simple finite element for the analysis of laminated plates, Comput. Struct. 44(3) (1992) S85-596.
A.K. Ghosh and S.S. Dey, Buckling of laminated plates-a
simple finite element based on higher-order theory, Finite Elem. Anal. Des.
I5 ( 1994) 289-302.
C.P. Wu and W.Y. Chen, Vibration and stability of laminated plates based on a local high order plate theory. J. Sound Vib. l77(4)
(1994) 503-520.
[37] J.S. Mona, C.M. Mota Soares and C.A. Mota Soares, Buckling behaviour of laminated composite structures using a discrete
higher-order displacement model, Composite Struct. 35( I ) (1996) 75-92.
[38] T.Y. Kam and R.R. Chang, Buckling of shear deformable laminated composite plates, Composite Struct. 22 (1992) 223-234.
[39] A.K. Noor and J.M. Peters, Finite element buckling and postbuckling solutions for multilayered composite panels, Finite Elem. Anal.
Des. I5 (1994) 343-367.
(401 A.W. Leissa. A review of laminated composite plate buckling, Appl. Mech. Rev. 40 (1987) 575-591,
[4l] A.W. Leissa, An overview of composite plate buckling, Composite Struct. Vol. 4, I.H. Marshall, ed. (Elsevier, London, I, 1987)
I. l-l .29.
[42] R.K. Kapania and S. Raciti, Recent advances in analysis of laminated beams and plates, Part 1: Shear effects and buckling, AIAA J. 27
(1987) 923-934.
[43] R.K. Kapania and S. Raciti, Recent advances in analysis of laminated beams and plates, Part 2: Vibration and wave propagation, AIAA
J. ?7(7) (1989) 935-946.
[44] A.N. Palazotto and S.T. Dennis, Nonlinear Analysis of Shell Structures (AIAA Education Series, Washington, DC, USA, 1992,)
[45] G.J. Simitses, Buckling of moderately thick laminated cylindrical shells: a review, Composites Part B, 27B(6) (1996) 581-587.
[46] H.M. Adelman and R.T. Haftka, Sensitivity analysis for discrete structural systems, AIAA J. 24(5) (1986) 823-832.
[47] R.T. Haftka and R.M. Adelman, Recent developments in structural sensitivity analysis, J. Struct. Optim. I (1989) 137-151.
[48] M. Zyczowski, Recent advances in optimal structural design of shells, Europ. J. Mech. A/Solids II (1992) 5-24.
[49] R. Grandhi, Structural optimization with frequency constraints-a
review, AIAA J. 31(12) (1993) 2296-2303.
[50] S. Abrate, Optimal design of laminated plates and shells, Composite Struct. 29 (1994) 269-286.
1511 S. Adali, A. Richter, V.E. Verijenko and E.B. Summers, Optimal design of hybrid laminates with discrete ply angles for maximum
buckling load and minimum cost, Composite Struct. 32 (1995) 409-415.
152
C.M.M.
Soares et al.
1 Comput.
[521S. Adali, M. Walker and V.E. Verijenko, Multiobjective
[53]
[54]
1551
[56]
[57]
[58]
[59]
[60]
[61]
[62]
[63]
[64]
[6.5]
[66]
[67]
[68]
[69]
[70]
[7l J
[72]
Methods
Appl.
Mech. Engrg.
149 (1997)
133-152
optimization of laminated plates for maximum prebuckling, buckling and
postbuckling strength using continuous and discrete ply angles, Composite Struct. 35 (1996) 117-130.
P.S. Frederiksen, Natural vibrations of free thick plates and identification of transverse shear moduli, in: P. Pedersen, ed., Optimal
Design with Advanced Materials (Elsevier Science Publishers, BV, Amsterdam, 1993) I3 I - 148.
P.S. Frederiksen, Advanced techniques for the identification of material parameters of thick composite plates, Proc. of ICCSTl I-First
International
Conference
on Composite Science and Technology,
S. Adali and V Verijenko, eds., Department of Mechanical
Engineering, University of Natal, Durban, South Africa (1996) 143-148.
A.L. Araujo, C.M. Mota Soares and M.J. Moreira de Freitas, Characterization
of material parameters of composite plate specimens
using optimization and experimental vibrational data, Composites Part B: Engineering, 27B(2) (1996) 185-191.
A.L. Aratijo, C.M. Mota Soares, M.J. Moreira de Freitas and P. Pedersen, Identification of mechanical properties of composite plate
specimens using a discrete higher-order displacement model and experimental vibration data, in: B.HX Topping, ed., Advances in
Analysis and Design of Composites, (Civil-Comp Press, Edinburgh, Scotland, 1996) 101-108.
B. Csonka, I. Kos& C.M. Mota Soares and C.A. Mota Soares, Shape optimization of axisymmetric shells using a higher-order shear
deformation theory, Struct. Optimiz. 9(2) (1995) 117-127.
V.M. Franc0 Correia, C.M. Mota Soares and C.A. Mota Soares, Design sensitivity analysis and optimal design of composite structures
using higher-order discrete models, Engrg. Optimization, to appear.
J.R. Vinson and R.L. Sierakowski, The Behavior of Structures Composed of Composite Materials (Martinus Nijhoff, Dordrecht, The
Netherlands,
1996).
O.C. Zienkiewicz, The Finite Element Method in Engineering Science, 3rd edition (McGraw-Hill,
London, 1987).
HV. Lakshminarayana
and S.S. Murthy, A shear-flexible triangular finite element model for laminated composite plates, Int. J. Numer.
Methods Engrg. 20 (1984) 591-623.
C.M. Mota Soares, V Franc0 Correia, H. Mateus and J. Herskovits, A discrete model for the optimal design of thin composite
plate-shell type structures using a two-level approach, Composite Struct. 30 (1995) 147-157.
K.L. Batoz, K.3. Bathe and L.W. Ho (1980)A study of three node triangular plate bending elements, Int. J. Numer. Methods Engrg.
(1980) 1771-1812.
K.J. Bathe and L.W. Ho, A simple and effective element for analysis of general shell structures, Comput. Struct. I3 (198 I ) 673-681.
A.P. Seyranian, E. Lund and N. Olhoff, Multiple eigenvalues in structural problems, Struct. Optimiz. 8 (1994) 207-227.
N. Olhoff, L.A. Krog and E. Lund, Optimization of multimodal structural eigenvalues, WCSMO-I, Proc. First World Congress of
Structural and Multidisciplinary
Optimization, N. Olhoff and G.I.N. Rozvany, eds. (Pergamon Press, UK, 1995) 701-708.
H.C. Mateus, H.C. Rodrigues, CM. Mota Soares and C.A. Mota Soares, Sensitivity analysis of thin laminated structures with a
non-smooth eigenvalue based criterion, WCSMO-I, Proc. First World Congress of Structural and Multidisciplinary
Optimization, N.
Olhoff and G.I.N. Rozvany eds. (Pergamon Press, UK, 1995) 689-694.
G.N. Vanderplaats, Numerical Optimization Techniques for Engineering Design: With Applications (McGraw-Hill
Inc., New York,
1994).
T.Y. Kam and M.D. Lai, Multilevel optimal design of laminated composite plate structures, Composite Struct. 31 (1989) 197-202.
A.V. Soeiro, C.A. Ant6nio and A.T. Marques, Multilevel optimization of laminated composite structures, Struct. Optimiz. 3 (1991)
69-78.
N.J. Pagan0 and S.J. Hatfield, Elastic behavior of multilayered bidirectional composites, AIAA J. IO (1972) 931-933.
A.K. Noor, Free vibrations of multilayered composite plates, AIAA J. I l(7) (1973) 1038-1039.
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement